Post on 22-Sep-2020
Credit Scoring and Mortgage Securitization: Implications for Mortgage Rates and Credit Availability
December 21, 2000
Andrea Heuson Associate Professor of Finance University of Miami Box 248094 Coral Gables, FL 33134 Aheuson@miami.edu (305) 284-1866 Office (305) 284-4800 Fax Wayne Passmore Assistant Director Federal Reserve Board Mail Stop 93 Washington, DC 20551 Wayne.passmore@frb.gov (202) 452-6432 Office (202) 452-3819 Fax Roger Sparks Associate Professor of Economics Mills College 5000 MacArthur Blvd. Oakland, CA 94613-1399 Sparks@mills.edu (510) 430-2137 Office (510) 430-2304 Fax
We wish to thank Steve Oliner, David Pearl, Tim Riddiough, Robert Van Order, Stanley Longhofer, and an anonymous referee for helpful comments on previous drafts of this paper. We take responsibility for all errors.
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Credit Scoring and Mortgage Securitization:
Implications for Mortgage Rates and Credit Availability
Abstract
This paper develops a model of the interactions between borrowers, originators,
and a securitizer in primary and secondary mortgage markets. In the secondary
market, the securitizer adds liquidity and plays a strategic game with mortgage
originators. The securitizer sets the price at which it will purchase mortgages
and the credit-score standard that qualifies a mortgage for purchase. We
investigate two potential links between securitization and mortgage rates. First,
we analyze whether a portion of the liquidity premium gets passed on to
borrowers in the form of a lower mortgage rate. Somewhat surprisingly, we find
very plausible conditions under which securitization fails to lower the mortgage
rate. Second, and consistent with recent empirical results, we derive an inverse
correlation between the volume of securitization and mortgage rates. However,
the causation is reversed from the standard rendering. In our model, a decline in
the mortgage rate causes increased securitization rather than the other way
around.
3
I. Introduction
This paper develops a model of the primary and secondary mortgage
markets. The primary market is competitive, consisting of numerous originators
and a continuum of borrowers with differing default probabilities. In the
secondary market, a monopolist sells mortgage-backed securities, which yield a
liquidity benefit, in exchange for mortgages offered by originators. The
monopolist/securitizer sets both the price for these mortgages and the credit-
quality standard that qualifies a mortgage for purchase. Although credit scoring
ensures that originators do not enjoy an information advantage over the
securitizer, they do enjoy a “first-mover advantage” in selecting which qualifying
mortgages to sell. The main purpose of the analysis is to shed light on how
securitization affects the interest rate paid by borrowers and the availability of
mortgage credit.
Historically, originators of residential mortgages have had two distinct
advantages vis-a-vis mortgage securitizers. First, originators had better
information about the creditworthiness of borrowers and their risks of default on
mortgages. Originators processed loan applications and followed trends in local
real estate markets, thereby acquiring knowledge about the riskiness of local
borrowers’ income streams and the market values of properties. Second,
originators had a first-mover advantage in the selection of mortgages to keep in
their portfolios. Each originator unilaterally chose which qualifying mortgages to
pass on to the securitizer.
4
With the recent advent of automated underwriting, much of the
informational advantage has disappeared. As the argument goes, computerized
credit scoring gives the securitizer more accurate and timely information about
borrower creditworthiness.1 On the other hand, the first-mover advantage
endures because originators still decide whether or not to securitize each
qualifying mortgage. Furthermore, the evidence suggests that mortgage
securitizers are aware of the originator’s first-mover advantage;2 such awareness
is a precondition for strategic interaction.
While credit scoring improves the quality of information, securitization
conveys an important benefit to mortgage originators (or lenders). By holding a
mortgage-backed security rather than the mortgage itself, lenders achieve
greater liquidity. A key question is whether this benefit gets passed on to
borrowers. Specifically, does the liquidity benefit of securitization translate into a
lower mortgage rate and/or greater access to credit? To investigate, we begin by
developing a baseline model of borrower and lender behavior in a competitive
mortgage market without mortgage securitization.
1 Somewhat paradoxically, however, automated underwriting can have a negative impact on securitizer profits, as shown in Passmore and Sparks (2000). 2 The chairman of Fannie Mae was quoted in a speech to mortgages bankers: “If the risk profile of mortgages you deliver to us differs substantially from the risk profile of your overall book of business, then we will have no choice but to believe we have been adversely selected.” Jim Johnson, as quoted in “Comment: Wholesale Lending Leaves Mortgage Out of the Loop,” American Banker, October 31, 1995.
5
Although the baseline model serves as a useful benchmark for
comparison, it does not adequately capture the institutional structure of U.S.
mortgage markets. Consequently, we extend the model by adding a mortgage
securitizer who behaves strategically. The extended model builds on work by
Passmore and Sparks (1996), who demonstrate that a mortgage securitizer can
reduce an originator’s screening of loans—thus reducing the volume of poorer-
quality mortgages passed on to the securitizer—by raising the interest rate
offered on the mortgage-backed securities that the securitizer swaps for
mortgages.
Several studies ascribe market benefits to asset securitization. In a paper
promoting the development of government-sponsored mortgage securitization,
Jones (1962) points to improved liquidity as a key effect. More recently, Black,
Garbade, and Silber (1981) and Passmore and Sparks (1996) argue that the
implicit government guarantee enhances liquidity.3 Within general asset markets,
Greenbaum and Thakor (1987) show that banks, by selling loans rather than
funding them through deposits, can provide a useful signal of loan quality. Hess
and Smith (1988) show that asset securitization is a means of reducing risk
through diversification. Boot and Thakor (1993) demonstrate that this
diversification may improve information. When assets are assembled in
portfolios, the payoff patterns that they yield are easier to evaluate because
diversification eliminates asset idiosyncrasies. Donahoo and Shaffer (1991) and
3 Gorton and Pennacchi (1990), Amihud and Mendelson (1986), and Merton (1987) show that there are trading gains associated with increased liquidity.
6
Pennacchi (1988) suggest that banks securitize assets in order to lower reserve
and capital requirements and thereby reduce financing costs.
Recent research also highlights several potential drawbacks to asset
securitization. Securitization may aggravate problems of asymmetric information
concerning the credit quality of loans. Passmore and Sparks (1996) emphasize
adverse selection, which gives originators an informational advantage over a
mortgage securitizer. Pennacchi (1988) stresses moral hazard, which arises
because the bank has less incentive to monitor and service loans after they are sold.
The present paper deviates from the earlier model of Passmore and
Sparks (1996) by assuming that the securitizer is well informed about credit
quality. The securitizer observes each mortgage applicant’s credit score, which is
a perfect signal of the applicant’s probability of not defaulting on a mortgage. In
addition, we allow the securitizer to choose both the interest rate offered on
mortgage-backed securities and the credit score standard that borrowers must
meet to qualify their mortgages for securitization.4 The underwriting standard
represents another tool that the securitizer may use to influence the credit quality
of the securitized mortgage pool. Finally, we introduce a continuum of no-default
probabilities.
Our primary finding is that mortgage securitization does not necessarily
lower the equilibrium mortgage rate. While mortgage securitization does alter the
placement of mortgages between the originator and the securitizer, it may leave
unchanged the cost of holding the marginal mortgage. In this case, the
4 `Calomiris, Kahn, and Longhofer (1994) allow lenders to set both the price and a cutoff standard.
7
originator’s unaltered marginal profitability condition determines the equilibrium
mortgage rate, and the presence of a securitizer offering a liquidity premium does
not affect that rate.
Our results also suggest a reinterpretation of recent empirical evidence
concerning a negative correlation between mortgage rates and the volume of
securitization. Researchers generally interpret this correlation as indicating
causation in that greater securitization reduces mortgage rates. Kolari, Fraser,
and Anari (1998), for example, conclude that an increase of 10-percent in the
proportion of mortgages securitized will decrease yield spreads on home loans
by approximately 20 basis points. Our model, in contrast, predicts that a decline
in mortgage rates causes the volume of securitization to rise, a reversal of
causation from previous interpretations.5
Throughout this paper, we abstract from the issues of prepayment risk and
mortgage insurance. Although these are significant features of U.S. mortgage
markets, we wish to focus on the liquidity premium and the assumption of credit
risk by a third party. Therefore, we assume that potential mortgage borrowers
differ from one another only in their probabilities of defaulting on a mortgage.
5 In addition to Kolari, Fraser, and Anari (1998), see Black, Garbade, and Silber (1981), who argue that the pass-through program sought to reduce yields on mortgages by improving their marketability. The growth of securitization coincides with the standardization of mortgage contracts and the creation of a national mortgage market during the 1980s; untangling these trends, some of which likely lower the cost of originating mortgage credit, is difficult. Todd (2000) finds that securitization is uncorrelated with mortgage rates but is inversely correlated with mortgage origination fees.
8
The next section of the paper presents a baseline model of borrower and
lender behavior in a competitive market with perfect information. Section III
presents the extended model, while section IV offers concluding remarks.
II. The Baseline Model: Without Securitization
Let the probability of a household not defaulting on a mortgage be
denoted by q , and assume that the q ’s are distributed according to a
continuous, differentiable, and single-peaked density function f (q) defined over
the line interval [0,1]. In both the baseline and extended models, we assume that
f (q) > 0 ∀q ∈(0,1) ,6 that the distribution of q s is public knowledge, and that each
mortgage applicant’s no-default probability is publicly observed. The key
difference is that in the extended model, a securitizer commits to purchasing all
mortgages meeting a credit-score standard.
Under the assumption of risk neutrality, a mortgage originator will offer a
mortgage to any applicant who incrementally adds to the lender’s expected profit.
Formally, an applicant will be offered a mortgage if and only if
qr + (1− q)rd ≥ rf , (1)
where q is the applicant’s probability of not defaulting on the loan, r is the
6 An example of a probability density function meeting these assumptions is the beta density
function: f (q) =Γ(α1 + α 2)Γ(α1)Γ(α2 )
q α1 −1(1− q)α 2−1
for 0<q <1 and f (q) = 0 otherwise.
9
mortgage rate received by the lender if the borrower does not default,7 rd is the
expected return to the lender if the borrower does default, and rf is the expected
return on an alternative investment. We assume rf > rd so that lenders do not
profit from mortgage default. Writing (1) as an equality and solving for r , we
obtain the lowest mortgage rate the lender is willing to accept as a function of the
applicant’s no-default probability:
rmin = rd +(rf − rd )
q. (2)
Equation (2) is the inverse supply function, which is decreasing in q and rd but
increasing in rf .
Now consider the decision facing a mortgage applicant. Denoting the
expected benefit to an applicant from possessing a mortgage by rb , and the cost of
defaulting on a mortgage by rc , then an applicant will apply for a mortgage if and
only if the benefit is at least as great as the cost, or
qr + (1− q)rc ≤ rb , (3)
7 We assume that the mortgage rate is uniform across borrowers even though they have different credit risks. To justify this assumption, we point to common practice. Originators typically make the mortgage rate contingent upon loan qualification but not upon the applicant’s actual credit score. See Leeds (1987). There are many plausible reasons why banks charge rates that are not fully contingent upon borrower credit risk. A uniform rate may be an efficient means of risk sharing between a risk-neutral lender and risk-averse applicants who are initially uninformed about their true default probabilities. This explanation is suggested by Calomiris, Kahn, and Longhofer (1994), p. 670. Another possible reason is that it may be costly to verify and enforce risk-based contracts, particularly when government regulators are trying to discourage discrimination in credit markets. For example, the courts may have difficulty sorting out whether an originator who charges a higher rate to a minority or low-income borrower is practicing discrimination or efficient risk pricing.
Even when risk-based pricing does occur, it generally takes form as a discontinuous step function rather than as a continuous function of the credit risk distribution. The lender divides applicants into risk categories, for example, low, medium, and high (but acceptable), and then charges a single rate within each category.
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where we assume that rb < rc , so borrowers do not expect to gain from default.
Setting (3) as an equality and solving for r , we may write the maximum mortgage
rate the applicant is willing to pay for the loan as:
rmax = rc −(rc − rb)
q. (4)
Equation (4) is the inverse demand function, which is increasing in q and rb but
decreasing in rc .8
To make mortgages mutually agreeable to some borrowers and lenders, we
assume that rb > rf . Similarly, to prevent default from being mutually beneficial, we
assume that default is expected to cost the borrower more than it benefits the
lender, i.e., rc > rd . Combining all of our assumptions on parameter values, we
have:
rc > rb > rf > rd . (5)
If there are many price-taking lenders, we may equate (2) and (4) to solve
for the market-equilibrium, no-default probability, q * , which is the probability of not
defaulting for the marginal borrower obtaining a mortgage:
q * = 1−(rb − rf )(rc − rd )
. (6)
8 By contrast, in Stiglitz and Weiss (1981), the maximum interest rate that borrowers are willing to pay decreases with the credit quality of borrowers. In their model, borrowers with higher default risk take greater investment risks that earn higher expected returns. Hence, these borrowers are willing to pay higher interest rates on loans. In our model, on the other hand, the rate of return on borrowed funds is constant across potential borrowers and not related to default probabilities (i.e., rb is independent of q ). Furthermore, we assume that default is sufficiently costly for borrowers so that those with higher risks of default (lower q ) have less willingness to pay for mortgages.
11
Note that (5) and (6) imply q * ∈(0,1). Substituting (6) into (2) or (4), we find the
market equilibrium mortgage rate9
r * = rd +(rf − rd )(rc − rd )(rc − rb + rf − rd )
, (7)
which from (5) implies that r * ∈(rd ,rc ). We may illustrate the market equilibrium by
graphing the inverse supply and demand functions, i.e., equations (2) and (4), as in
figure 1. The graph shows the equilibrium mortgage rate r * and the equilibrium
marginal no-default probability q * . Households obtaining mortgages are those with
no-default probabilities contained by the line interval [q *,1]. Next, we extend the
model to encompass securitization in the secondary market.
9 This equilibrium concept implies that originators earn positive profits in equilibrium. They earn zero profit on the marginal borrower and positive profit on the inframarginal borrowers, since each borrower pays the same mortgage rate. Although we have not modeled a process in which these profits are competed away, there are several possibilities. Competition in providing loan services, in obtaining access to land for branches, in advertising, and in general rent-seeking behavior may cause “fixed costs” (i.e., costs that are independent of borrower credit risk) to rise high enough to soak up any positive profits.
12
r c
r *
r d
Inverse Demand
Inverse Supply
1
households obtaining mortages
r f
r b
Figure 1: Equilibrium in the Baseline Model
r max = r c − ( r c − r b )
q
q * q
r min = r d + ( r f − r d )
q
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III. The Extended Model: Credit Scoring and Securitization
A. Model Structure
We now modify the baseline model by adding a mortgage securitizer10
who is willing to bear the credit risk of mortgages that meet certain conditions.
We assume that the securitizer has the same information as do mortgage
originators about the default risks of borrowers, but originators (whom we shall
also refer to as banks) choose which mortgages to hold in their own portfolios
and which ones to securitize.
Suppose that banks and the securitizer make sequential decisions over
three periods. In period 1, the securitizer specifies two contractual terms for
swapping a mortgage for a mortgage-backed security (MBS): an interest rate rs
offered on the MBS, and a credit standard q that the mortgages must meet in
order to qualify for securitization.11 Also during this period, banks receive loan
applications and information about the applicants’ creditworthiness. This
information includes applicants’ loan default probabilities, which are disclosed to
both banks and the securitizer.
Using information on borrower default probabilities, a bank in period 2
attempts to maximize profits by taking action on submitted mortgage
10 By assuming a monopolist securitizer, our model abstracts from the actual market structure for fixed-rate conforming loans in the United States, where Fannie Mae and Freddie Mac are duopolists. In our judgment, this abstraction represents a reasonable trade-off between realism and tractability. However, an extension to duopoly would be a promising path for future modeling. 11 The securitizer simply issues pass-through securities that are immediately swapped with the banks that originated the mortgages. Thus, the guaranteed interest payments from the securitizer to the originator are self-funded out of the pool of securitized mortgages.
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applications. Some mortgages are granted and held in the bank’s portfolio;
others are granted and immediately traded for mortgage-backed securities; and
the remaining applications are rejected.
In period 3, borrowers either default on their mortgages (in which case the
rate of return to the mortgage holder is rd ) or pay them back in full. At the end of
this period, all parties receive their payoffs.
The sequential game played between the securitizer and the bank is
depicted in figure 2. The securitizer first chooses q and rs . Then, both bank and
securitizer observe applicant default probabilities. For conforming mortgages,
the bank decides either to sell the mortgage to the securitizer (securitize) or hold
the mortgage in its own portfolio (keep). For mortgages that do not meet the
securitizer’s standards, the bank’s choice is either to keep the mortgage in its
portfolio or to reject the application (reject). Finally, the borrower either defaults
(d) or does not default (nd) on the loan, and the bank and securitizer receive the
payoffs.
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sets
conforming nonconforming
keep securitize keep reject
nd d nd d dnd nd d
Securitizer
publicly revealed
BankBank
Payoffs:
BankSecuritizer
r
0
rd
0
rs +δ rs +δr − rs rd − rs
rd
0
r
00
00
0
q ,rs{ }
q ≥ q q < q
q
Figure 2: The Game Between the Securitizer and the Bank
16
The return on a defaulted mortgage is rd , while the return on a
nondefaulted mortgage is r . The total return to the bank from securitizing a
mortgage is rs +δ , where δ > 0 is the value of liquidity to the bank from holding a
mortgage-backed security as opposed to the mortgage itself, or the “liquidity
premium.”
Let the number of households be denoted by N > 0 . Following the
baseline model, we assume that household probabilities of not defaulting on a
mortgage are distributed according to the density function f (q) defined over the
interval [0,1]. The distribution of q s is public knowledge, and each household
knows its own no-default probability, which is a random draw from the density.
Given its draw, a household decides whether to apply for a mortgage. As in the
baseline model, we assume that households are rational, applying for mortgages
only if the expected benefits exceed the expected costs of doing so. A
household applies only if:
rb ≥ qr + (1− q)rc , (8)
which implies
q ≥rc − rb
rc − r≡ γ , (9)
where γ is the lower bound on the distribution of no-default probabilities for
applicants.12 Households with no-default probabilities lower than γ do not apply
for mortgages.
12 We continue to assume that rc > rb ; also note that a nontrivial equilibrium requires rb > r ,
which from (9) implies γ ∈ (0,1) .
17
We define a conforming loan (that qualifies for securitization) as one
whose credit score is at least as great as a threshold q set by the securitizer.13
Since credit scores in this model are perfect indicators of no-default probabilities,
it follows that applications with q < q do not qualify for securitization. The
proportion of households that qualify for securitization is given by
[1−F (q )] for q ∈[0,1], which is decreasing in the credit standard q .
B. The Bank’s Problem
The bank will either grant the mortgage and keep it in portfolio, grant the
mortgage and immediately trade it for a mortgage-backed security, or reject the
application. For the securitizer to induce banks to originate and securitize any
mortgages, the return from securitization must be at least as great as the
alternative return; otherwise, banks will prefer their alternative investment.
Consequently, we impose the parameter restriction: rs +δ ≥ rf .
Consider a bank’s choice of which action to take with a mortgage
application. The bank rejects the loan application if
q <(rf − rd )(r − rd )
≡ qmin and q < q . (10)
The first inequality in (10) implies that holding the mortgage is not profitable for the
bank, while the second inequality says that the mortgage does not qualify for
securitization. Choice (10) and subsequent choices are illustrated in figure 3.
The bank accepts and holds the mortgage if
13 We analyze the determination of q later.
18
q >(rs + δ − rd )
(r − rd )≡ q’ (11)
or if
q ≥ qmin and q < q . (12)
Condition (11) implies that it is more profitable for the bank to hold the mortgage
than to securitize it, whereas (12) says that the bank finds the mortgage
profitable to hold but the mortgage does not qualify for securitization.
Finally, the bank accepts and securitizes the mortgage if
q ≤ q ’ and q ≥ q . (13)
In case (13), the mortgage is profitable for the bank to securitize and the
mortgage qualifies for securitization. In this instance, the securitizer receives
mortgages that are conforming but not sufficiently attractive to be retained by the
bank. Conditions (11) and (13) reflect the cherry picking that results from the
bank’s first-mover advantage. Of the qualifying mortgages, the bank retains the
higher-quality ones, satisfying (11), and sells the lower-quality ones, satisfying
(13).
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Note that if q ≤ qmin < q’ , then the marginal mortgage is securitized. We next
analyze the securitizer’s problem of choosing the interest rate rs and credit
standard q that maximize profits.
C. Securitizer Behavior
Suppose the securitizer, while taking r and rf as given, chooses q and rs
to maximize profits:14
max(q ,r s )
πS = N f (q)q
q’
∫ qr + (1−q)rd − rs[ ]dq , (14)
14 By assuming that the mortgage rate is taken as given, we are treating the securitizer as a price-taker in the primary mortgage market but a monopolist in the secondary market. It is straightforward to establish that the solution to the securitizer’s problem is an interior solution unless further parameter restrictions are imposed.
0 1
f (q)
γ q q’
Figure 3: Partitioning the Distribution of Borrowers
q
qmin
reject accept and bank holds
20
where the term inside the square brackets of (14) is the securitizer’s expected
profit from a securitized mortgage with no-default probability q .15
Assuming positive solutions for q and rs , we compute the following first-
order conditions:
∂πS
∂q = −Nf(q )[q r + (1−q )rd − rs ] = 0 and (15)
∂πS
∂rs
=N
(r − rd )f (q ’) q ’r + (1− q’ )rd − rs[ ]−N f (q )dq
q
q ’
∫ = 0 , (16)
where we recall from (11) that q’=(rs + δ − rd )
(r − rd ). By prior assumptions, N,f (q ) > 0 ,
which imply in (15) that:
q =(rs − rd )(r − rd )
. (17)
Equation (17) shows that q is set so that the securitizer would earn zero profit by
securitizing the marginal qualifying mortgage, which is a standard condition for
profit maximization.
Using (11) and (17), we may rearrange (16) to obtain:
frs + δ − rd
r − rd
δ(r − rd )
= Frs +δ − rd
r − rd
− F
rs − rdr − rd
, (18)
15 The securitizer here is assumed to have the status of a government-sponsored enterprise, which can convert one dollar of mortgages into one dollar of assets and still create the liquidity premium. In contrast, a purely private-sector securitizer would need to purchase credit enhancements or create a senior/subordinated debt structure to minimize investors’ credit risk concerns and create a liquidity premium. This latter case is discussed at length in Passmore, Sparks, and Ingpen (2000).
21
an equation with one endogenous variable, rs , because the solution for q comes
from (15). Equation (18) has a standard economic interpretation. The guaranteed
rate rs is set so that the marginal net revenue from raising rs (left side) equals the
marginal cost to the securitizer (right side). Increasing rs raises q’ by the factor
1(r − rd )
and induces banks to securitize additional mortgages, thereby generating
marginal net revenue of frs + δ − rd
r − rd
δ .16 The right side of (18) represents the
marginal cost of raising rs because the securitizer pays a higher price for all
securitized mortgages.
To derive the key comparative static results for the securitizer’s choice
problem, we briefly examine the second-order conditions associated with (14).
Substitution of (17) into (16) eliminates q as an explicit choice variable and allows
us to state second-order conditions applicable to the single choice variable rs .
After making this substitution, we differentiate the resulting expression with
respect to rs . Then, we impose the condition that this derivative be less than
zero:
f ’ δ(r − rd )
− f + f 0 < 0 , (19)
16 The marginal net revenue from raising rs is MR = f (q’ )[q ’r + (1− q’ )rd − rs ]. Substitute
(11) into this expression to get MR = f (q’ )(rs +δ − rd )
(r − rd )(r − rd ) + rd − rs
= f (q ’ )δ .
22
where f ’ denotes the first derivative of the pdf evaluated at rs + δ − rd
r − rd
, f is the pdf
evaluated at rs + δ − rd
r − rd
, and f 0 is the pdf evaluated at rs − rd
r − rd. In section A of the
appendix, we derive another second-order condition:
f ’ < 0. (20)
Thus, (19) and (20) are second-order conditions for the choice of rs to maximize
the securitizer’s profits.
We now address a comparative static question: How does the securitizer’s
choice of MBS rate rs respond to movements in the market interest rate r ?
Taking the differential of (18) with respect to rs and r , we find
f ’ δ(r − rd )
− f + f 0
drs −
(rs − rd )(r − rd )
f ’ δ(r − rd )
− f + f 0
+ f ’ δ 2
(r − rd )2
dr = 0 . (21)
Since the term inside the two sets of square brackets in (21) is negative by (19)
and f ’ < 0 by (20), we have
∂rs
∂r> 0 . (22)
That is, the securitizer responds to increases in the mortgage rate by raising the
MBS rate. Using (21) and (22), we may differentiate (17) to find
∂q ∂r
=
∂rs
∂r(r − rd ) − (rs − rd )
(r − rd )2 =f ’δ 2
(r − rd )3
1
f ’δ(r − rd )
− f + f 0
> 0 , (23)
which shows that the securitizer raises the credit standard in response to
increases in the mortgage rate. The intuition for this result is appealing. Knowing
23
that a rise in the mortgage rate induces originators to retain some additional high-
quality mortgages (i.e., q’ declines), the securitizer responds by raising the MBS
rate to counter this increased cherry picking. But the higher MBS rate raises the
securitizer’s costs of purchasing mortgages, making some low-quality mortgages
unprofitable to purchase. The securitizer raises the credit standard to exclude
those mortgages from the conforming pool.
D. The Supply of Credit Under Securitization
To investigate the supply side of the primary market, we now graph q as a
function of r . Taking limits of (23), we find: limr →rd
∂q ∂r
= ∞ and limr →∞
∂q ∂r
= 0 .
Consequently, the graph of q (r ) from (17), taking into account the dependence of
rs on r from (18), has the general shape shown in figure 4 (though it is not
necessarily everywhere strictly convex to the origin). The function q (r ) shows the
securitizer’s willingness to accept mortgages for securitization as a function of r .
Another aspect of mortgage supply in the primary market is the bank’s
willingness to securitize mortgages as opposed to rejecting them and earning rf .
For this securitization choice not to lower the bank’s profits, the weak inequality
rs +δ ≥ rf (24)
must hold. To graph (24) in (q,r ) space, we substitute (17) into (24) and obtain17
17 Equation (17) gives the securitizer’s choice of q as a function of rs . Solving (17) for rs , we
obtain the maximum rs the securitizer would pay for credit standard q : rs = q (r − rd ) + rd . Substitution of this expression into (24) yields (25).
24
q ≥rf − δ − rd
r − rd≡ q , (25)
which shows the condition for the bank to choose securitization over rejection of
the mortgage application. The graph of q(r) is shown in figure 4.
The final component of market supply derives from the bank’s willingness
to hold mortgages rather than reject them, as expressed in (2). Solving (2) for q ,
we have
qmin =rf − rdr − rd
, (26)
which is interpreted as the minimum no-default probability that causes the bank to
hold rather than reject the mortgage. By comparing (25) and (26), we see that
q(r) lies δ
r − rd
units to the left of qmin(r ) . Through this mechanism, the liquidity
premium affects supply conditions in the primary market.
For a mortgage to be securitized, both the bank and the securitizer must be
willing to exchange the mortgage (with its risk of default) for payment of a
guaranteed rate rs . At any mortgage rate r , the willingness of both parties to
securitize the mortgage is given by the right envelope of q (r ) and q(r) , shown in
bold, or the “short side” of the market. The bold segments of q (r ) and q(r) thus
show the marginal securitized mortgage as a function of r .
25
We derive the inverse supply function by finding the set of lowest mortgage
rates for which a mortgage is offered and held (by either the bank or the
securitizer). For q ≥ ˜ q , the lowest rate is given implicitly by the q(r) function. In
that range of no-default probabilities, the securitizer is willing and able to hold
mortgages at a lower mortgage rate than are banks. In addition, banks are willing
to securitize these mortgages rather than reject them. On the other hand, for
q < ˜ q , the lowest mortgage rate is given implicitly by qmin(r ). In the range q < ˜ q ,
the securitizer and the bank do not find a mutually agreeable (rs,q ) combination
q
r q ( r )
Figure 4: Components of Supply in the Extended Model
q min ( r )
˜ q
q ( r )
26
for securitizing the marginal mortgage. Essentially, the securitizer balks at the
idea of holding high-risk mortgages (with no-default probabilities q < ˜ q ).
Let r −1(q) be defined implicitly by (25), and recall that rmin is given by (2). The inverse
supply function, rsupply , is then rmin for q < ˜ q and r −1(q) for q ≥ ˜ q , as shown in figure 5. The
discontinuity at ˜ q shows that securitization lowers the marginal cost of supplying mortgages
to borrowers with good credit risk (in the range q ≥ ˜ q ). However, securitization has no effect
on the supply of mortgages to borrowers with poor credit risk (in the range q < ˜ q ). The size of
q
r
Figure 5: Inverse Supply in the Extended Model
˜ q
{ δ ˜ q
=
r min
r − 1 ( q )
r supply
27
the discontinuous jump from rmin to r −1(q) depends directly on the size of the liquidity premium
and is equal to δ˜ q
.18
E. Market Equilibrium
In the extended model, the presence of a securitizer does not alter the
willingness of potential borrowers to pay for mortgages. As in the baseline
model, the maximum mortgage rates that households are willing to pay depend
on the cost of default, the benefit of owning a home, and the households’
probabilities of not defaulting on the loan. Since these parameters are not
altered by introducing a securitizer, the inverse demand function for mortgages
remains unchanged between the baseline and extended models, and is given by
(4). We have already shown how the presence of a securitizer affects the
inverse supply function.
In an equilibrium of the extended model, the securitizer’s choices of q and rs
must satisfy (15) and (16). Additionally, the mortgage market must be in equilibrium:
rsupply = rmax , (27)
where rmax is given by (4). When these three conditions are satisfied, they jointly
determine a market equilibrium (ˆ r , ˆ q , ˆ q , ˆ r s ) , consisting of a mortgage rate, marginal
no-default probability on an originated loan, credit standard, and MBS rate. We
18 This is derived from (2) and (25) solved for r . Substitute ˜ q = q = q into both expressions to
solve for the vertical discontinuity at ˜ q .
28
are interested in whether the equilibrium mortgage rate of the extended model ˆ r
is less than the equilibrium rate for the baseline model r * .
Two possible cases are depicted in figures 6 and 7.19 In figure 6, the
inverse demand function rmax intersects the inverse supply function rsupply to the
right of ˜ q . A new equilibrium emerges at B = ( ˆ q , ˆ r ), and it involves a lower
mortgage rate and greater volume of mortgages than at point A , the equilibrium
without securitization. In this case, a portion of the liquidity premium is
transferred to borrowers, and borrower surplus increases by area ABC. In the
new equilibrium, more borrowers qualify for loans, and all borrowers obtain loans
at a lower mortgage rate. As the graph shows, these benefits are more likely to
occur when rmax is relatively low. Furthermore, the size of the impact on the
mortgage rate is directly related to the size of δ .
To gain additional insight, we include in figure 6 the graphs of q (r ), from
(17), and q’ (r ) = q (r ) +δ
(r − rd ), from (11) and (17). At the new equilibrium, the
bank securitizes mortgages in the interval [ ˆ q ,q’ ] and holds mortgages in the
interval (q ’,1] . In this case, securitization conveys benefits to borrowers because
the marginal mortgage is securitized.
19 A third possible case arises if the inverse demand function passes through the discontinuous portion of the inverse supply curve. Analysis of this case would require refining the notion of equilibrium used in the paper.
29
Figure 7 illustrates the case in which securitization has no effect on the
mortgage rate and volume of mortgages since the function rmax intersects the
inverse supply curve to the left of ˜ q . In this case, the equilibrium remains at point
E (q* ,r * ) because the demand for mortgages is high. High-risk borrowers (with
no-default probabilities below ˜ q ) are willing to pay mortgage rates that induce the
banks to hold the marginal mortgage rather than securitize it. Hence, mortgage
supply conditions are unchanged at the margin.
At the equilibrium (q* ,r * ), the bank holds in its portfolio the marginal
mortgage and all mortgages in the interval [q *,q ). The bank securitizes
mortgages in the interval [q ,q’ ] , while keeping those in the interval(q ’,1] . The
q
r
Figure 6: Securitization Lowers the Mortgage Rate and Raises the Volume of Mortgages
˜ q
Inverse Supply Inverse Demand r max
A
B
q * ˆ q
r *
ˆ r C
q ( r ) q ' ( r )
q '
securitized
1
held by bank
r supply (in bold )
r min
30
securitizer’s impact in this case is simply to securitize mortgages that previously
were held in the banks’ portfolios, but none of the liquidity premium is passed on
to mortgage borrowers. Notably, the equilibrium in figure 7 does not involve
securitization of the marginal accepted mortgage.20
The results illustrated in figures 6 and 7 provide some clues about the
conditions under which securitization is likely to affect the mortgage rate and
access to mortgage credit. In figure 6, the demand for mortgage credit is
relatively low and securitization lowers the mortgage rate. Since the marginal
borrower is a good credit risk, the securitizer is willing to securitize that mortgage
and pass through the liquidity benefit to all borrowers. Conversely, in figure 7, the
demand for credit is high and securitization has no effect on the mortgage rate
because the marginal borrower is a poor credit risk.
F. Risk-Based Mortgage Rates
The results shown in figures 6 and 7 are derived under the assumption
that borrowers pay a uniform mortgage rate. However, borrowers in U.S.
markets sometimes pay rates that vary with risk classification (defined by credit-
score intervals), with borrowers in higher-risk classes paying higher rates. This
raises the question of how our results would be affected if banks in the model
charged rates that are similarly contingent on borrower risk.
20 Empirically, this seems to be the case. A potentially fruitful empirical approach would be to test whether the marginal mortgage is securitized over various segments of the business cycle, thereby providing insight into the conditions under which securitization may, according to our model, be effective in lowering mortgage rates and expanding loan access.
31
Consider the limiting case in which borrowers negotiate a continuum of
mortgage rates along the rmin function (without securitization) and the rsupply
function (with securitization), as depicted in figures 6 and 7.21 That is, each
borrower obtains a mortgage at the lowest interest rate banks are willing to offer
21 A pure continuum of prices does not appear to exist in the market. Instead, there are rate clumps. Thus, our limiting case gives the greatest possible leeway for securitization to affect the primary market, and the actual effects are likely to be less.
q
r
Securitization Has No Effect on the Mortgage Rate and Volume of Mortgages
˜ q
Inverse Supply
Inverse Demand r max
E
q ( r ) q ' ( r )
1 q q '
held by bank securitized held by bank
r supply
ˆ r = r *
ˆ q = q *
Figure 7:
r min
(in bold)
32
for no-default probability q .22 How does this pricing scheme affect the impact of
securitization on mortgage rates and credit availability?
There is no effect on credit access because the equilibrium conditions for
the marginal borrower remain unchanged. Under the stated scheme for risk-
based pricing, the equilibrium volume of mortgages remains at q * (where
rmin = rmax ) and ˆ q (where rsupply = rmax ), as shown in figures 6 and 7.
Consequently, securitization expands credit access only for the conditions of
figure 6 but not those of figure 7.
Now consider the impact on mortgage rates. In figure 6, securitization
lowers mortgage rates for all borrowers by shifting downward the continuum of
risk-based rates. Initially, rates are given by the portion of the rmin function to the
right of point A. Then, securitization causes the rate structure to shift downward
to the portion of the rsupply function to the right of point B.
In figure 7, securitization has an impact on risk-based rates for some
borrowers but not others. Borrowers with no default probabilities in the interval
( ˜ q ,1] benefit from a downward shift in their rate structure. But higher-risk
borrowers in the interval [ ˆ q , ˜ q ) reap no benefits from securitization. Their rate
structure remains the portion of the rsupply function between ˆ q and ˜ q .
The discussion above indicates that even the most finely-graduated
scheme for risk-based pricing does not fundamentally alter our results on the rate
impacts of securitization. With risk-based rates, securitization confers rate
22 This pricing scheme yields mortgage rates that vary continuously and inversely with credit scores (and no-default probabilities) while causing bank profits to be zero.
33
reductions to a group of lower-risk borrowers.23 However, securitization may not
reduce rates for higher-risk borrowers and therefore may not affect the prevailing
mortgage rate.24 Further, the risk-based scheme does not modify the effects of
securitization on credit availability. Next, we investigate several properties of
equilibrium with a uniform mortgage rate.
G. Propositions
Proposition 1 below identifies conditions under which securitization does
not affect equilibrium.
Proposition 1: The case in which securitization has no effect on the
equilibrium mortgage rate is more likely to arise for larger values of rb and
smaller values of rc and δ .
Proof: See section B of the appendix.
Proposition 1 tells us that the liquidity premium does not pass through to
borrowers in the form of a lower mortgage rate when mortgage demand is high
and/or the liquidity premium is low. The next proposition highlights the
securitizer’s choice of rs as pivotal in determining whether the liquidity premium
passes through to borrowers in the form of a lower mortgage rate.
23 These benefits are diluted if rates are a step function of credit-score intervals. 24 By ‘prevailing’ mortgage rate we mean the rate advertised by banks. This rate generally applies to the marginal borrower, while inframarginal borrowers are sometimes able to negotiate lower rates.
34
Proposition 2: Securitization will either increase the volume of mortgage
loans and lower the mortgage rate, in which case rs < rf , or it will leave both the
volume of loans and interest rate unchanged, in which case rs > rf .25
Proof: If securitization does affect equilibrium, then q < qmin because the
marginal mortgage is securitized. Using (17) and (10), we obtain
q < qmin ⇒ rs < rf . The proof for the case of no effect follows similarly. QED
Proposition 2 suggests an empirical test to indicate whether securitization
affects mortgage-market outcomes. Time periods in which the guaranteed rate
exceeds than the banks’ cost of funds (i.e., rs > rf ) are only consistent with
securitization having no effect (figure 7). On the other hand, rs < rf is only
consistent with securitization having some effect (figure 6).
Proposition 2 tells us that mortgage rates are unaffected by securitization
ifrs > rf . If one compares interest rates on mortgage-backed securities to the cost of
funds for banks, this inequality appears to hold. However, a more appropriate
comparison would adjust for the prepayment risk embedded in mortgage-backed
securities and then compare the "option-adjusted spread" to the banks’ cost of funds.26
Even with this adjustment, problems in measuring the marginal cost of
bank funds make it difficult to determine whether rs exceeds rf . For smaller
25 A positive liquidity premium makes it possible to have rs < rf in equilibrium. However, rs ≥ rf − δ is required for securitization in equilibrium. 26 In addition, the mortgage-backed security might provide diversification benefits that are not considered here.
35
banks, the marginal cost of funds might be best measured using the yield on
uninsured deposits, whereas for larger banks, the marginal cost of funds would
be better measured by subordinated debt. Regardless, such measurements are
complex and beyond the scope of this paper.
The potential ineffectiveness of securitization in lowering mortgage rates
raises a puzzling issue. Numerous empirical studies find an inverse correlation
between the volume of mortgages securitized and mortgage rates. This
evidence is frequently construed to imply that increases in securitization reduce
mortgage rates. Based on this interpretation, it is tempting to infer that, despite
the theoretical possibility of ineffectiveness, the empirically-relevant case is
securitization having some effect on mortgage rates. To address this issue, we
now analyze how the volume of securitization changes with the mortgage rate for
the case in which securitization has no effect on the mortgage-market
equilibrium.
Proposition 3: If securitization has no effect on the mortgage-market
equilibrium, then the volume of securitization varies inversely with the mortgage rate.
Proof: If securitization has no effect, on equilibrium, then the volume of
securitization is given by27
V = N f (q )dqq
q ’
∫ . (28)
27 For the case in which securitization does affect equilibrium, as shown in figure 6, the volume of securitization is not necessarily inversely related to the mortgage rate. In this instance, the
volume of securitization is determined by integrating from q to q’ , both of which are decreasing in r .
36
We have already shown that q varies directly with r in (23). A rise in r induces
the securitizer to raise q , which tends to decrease the volume securitized. But
we also need to investigate how q’ varies with r . Differentiating (11) and using
(23), we find28
∂q ’∂r
=
∂rs
∂r(r − rd ) − (rs + δ − rd )
(r − rd )2 =∂q ∂r
−δ
(r − rd )2 < 0, (29)
indicating that q’ declines with r. A rise in r thus causes q to increase (27) and
q’ to decrease (29), thereby decreasing the proportion of mortgages securitized
as the gap between q’ and q shrinks. The proposition is proved by substituting
(23) and (29) into the derivative of (28) with respect to r . Differentiating (28), we
have ∂V∂r
= Nf (q’ )∂q’∂r
− Nf(q )∂q ∂r
< 0, by (23) and (29). QED
The proof of proposition 3 demonstrates that changes in the mortgage rate
affect the securitizer’s choice of MBS rate, which in turn affects the upper and
lower bounds of no-default probabilities in the securitized pool, q’ and q .
However, notice that the gap between these probabilities (q ’−q ) is determined
by the securitizer’s choice of q . From (11) and (17), q’−q =δ
(r − rd ), showing
that the interval length of no-default probabilities (credit scores) in the securitized
pool varies with parameters exogenous to the securitizer. For example, a rise in
28 The sign of (29) is derived by substituting (23) into (29), simplifying, and then using the result from section C of the appendix that at the optimum f > f 0
.
37
the liquidity premium induces the securitizer to expand the gap between highest
and lowest credit scores in the pool; a rise in the mortgage rate has the opposite
effect.
Given the choice of q from (17), we now illustrate in figure 8 the
securitizer’s profit-maximizing choice of rs as characterized in (18).
Geometrically, profit maximization involves setting rs , which determines both q’
and q , to maximize the slope of the line segment from point A to point B, the
hypotenuse of a right triangle with fixed base length AC =δ
(r − rd ).29 Intuitively,
the securitizer chooses rs to maximize the volume of securitization, F (q’ ) − F(q ) ,
given the credit-score range for the securitized pool, δ
(r − rd ), where that range is
constrained by cherry-picking incentives. To use an analogy, the securitizer sets
rs to maximize the “catch” of securitized mortgages, given that the size of the
“net” is constrained.
On the basis of figure 8, it is straightforward to show the comparative
static effect stated in proposition 3. As r rises, the base length AC contracts.
The securitizer responds by raising rs , which raises q and lowers q’ . (Though
not shown in figure 8, points A and B would move closer together, with a
tangency between their new cord and the cdf at point B.) Since the changes in r
29 Further explanation is provided in section C of the appendix.
38
and rs cause q to increase and q’ to fall, the volume of securitization,
F (q’ ) − F(q ), falls.
The strategic interaction that lies behind proposition 3 is again cherry picking
and its mitigation. As the mortgage rate rises, so does the expected return from
holding mortgages, and the securitizer anticipates correctly that originators will want
to hold onto additional high-quality mortgages. That is,q’ declines. Raising the MBS
rate helps boost q’ back toward its original level. However, it does not pay for the
securitizer to undo all of the extra cherry picking; thus, on net, q’ declines. At the
same time, the securitizer’s profitability weakens at the opposite end of the credit
spectrum, where the solution is to elevate the credit standard q .
39
IV. Conclusion
In this paper, we analyze two models of the residential mortgage market.
The baseline model is constructed under the assumptions of perfect information, a
large number of borrowers and lenders, and the absence of a secondary market.
Not surprisingly, it yields results characteristic of a competitive market. To add
realism, we extend the model to include the presence of a securitizer who behaves
strategically toward mortgage originators.
Figure 8: The Securitizer’s Profit-Maximizing Choice of
q
F(q)F (q)
q’=(rs +δ − rd )
(r − rd )q =
(rs − rd )(r − rd )
δ(r − rd )
slope of curve at B =
slope of cord AB =
A
B
C
F (q’ ) − F(q )δ
(r − rd )
f (q’ )F (q )
F (q’)
F (q )
rs
40
The securitizer offers lenders additional liquidity (in the form of a mortgage-
backed security) and a guaranteed interest rate on a mortgage-backed security in
exchange for the risky return from individual mortgages. But the securitizer,
aware of lenders’ incentives in selecting mortgages for securitization, sets the
MBS rate and an underwriting standard on the mortgages it will accept, in order to
mitigate the cherry-picking problem. The “liquidity premium” from securitization
acts as a subsidy to originators on their inframarginal mortgage borrowers, but it
does not necessarily alter the opportunity cost of serving the marginal borrower.
Yet this marginal cost determines the equilibrium mortgage rate at the point of
intersection with the borrowers’ marginal benefit function. This theoretical result
suggests that the liquidity premium from securitizing mortgages may have little or
no effect on mortgage rates.
The model sheds light on the economic conditions under which
securitization is likely to have an impact on mortgage rates and access to
mortgage credit. If the demand for credit is relatively high in the model, then
securitization has no effect on the mortgage rate and loan volume. If conditions
are at the opposite extreme, then securitization lowers the mortgage rate and
improves credit access. These results suggest that securitization may exacerbate
fluctuations in mortgage rates, lowering rates only when they would otherwise be
low.30
Our work also suggests that securitization may reduce the impact of a
downturn in the demand for mortgages. If conditions are such that securitization
30 This implication is empirically testable.
41
does not impinge on the mortgage rate, then a decrease in loan demand may
cause a large drop in the mortgage rate (due to the discontinuity of supply). In
this instance, loan volume is contracted by less than it would be in a competitive
market without a securitizer.31 It would be interesting to test for this effect
empirically.
Our model carries an important implication for empirical work on the
relationship between securitization and mortgage rates. The negative correlation
between mortgage rates and the volume of securitized mortgages may be the
result of causation running from the mortgage rate to securitization volume, rather
than the other way around. Here, empirical tests of causality would provide
valuable evidence on the veracity of our model’s implications.
Our work raises questions about whether mortgage-market participants can
adapt their contractual relationships for mutual gain. For example, are there
contracting mechanisms for removing the barrier that sometimes prevents the
liquidity premium from being passed through as a lower mortgage rate for
borrowers? One contracting approach would be to attack the cherry-picking
problem, which derives from the originator’s first-mover advantage in selecting
mortgages to keep in portfolio. In the setting of a repeated game, the securitizer
could use future rewards, sanctions, and/or promises of repeat business to
discourage originators from cherry picking the highest-quality mortgages. But,
while such strategies would improve outcomes for the securitizer, they would not
necessarily affect the conditions for supplying the marginal borrower. Another
31 One empirical study suggests that FNMA had a countercyclical effect on the mortgage market during the 1980s. See Kaufman (1988).
42
approach would be for the securitizer and bank to offer risk-related guaranteed
rates and mortgage rates, respectively. But such contingent contracts may not
bring about efficient risk sharing and may complicate policies for deterring lending
discrimination.
43
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44
Merton, Robert C. “A Simple Model of Capital Market Equilibrium with Incomplete Information.” Journal of Finance 42 (July 1987), 483-510. Passmore, Wayne and Roger Sparks. “Automated Underwriting and the Profitability of Mortgage Securitization,” Journal of Real Estate Economics 28 (2000), 285-305. Passmore, Wayne and Roger Sparks, and Jamie Ingpen. “GSEs, Mortgage Rates and Mortgage Securitization,” unpublished manuscript (January 2000). Passmore, Wayne and Roger Sparks. “Putting the Squeeze on a Market for Lemons: Government-Sponsored Mortgage Securitization.” Journal of Real Estate Finance and Economics 13 (1996), 27-43. Pennacchi, George G. “Loan Sales and the Cost of Bank Capital.” The Journal of Finance 43 (June 1988), 375-396. Stiglitz, Joseph, and Andrew Weiss. “Credit Rationing in Markets with Imperfect Information.” American Economic Review 71 (June 1981), 393-410. Todd, Steven. “The Effects of Securitization on Consumer Mortgage Costs.” unpublished manuscript (January 2000).
45
Appendix
A. Derivation of f ’ < 0
To satisfy second-order conditions for a maximum, the quadratic associated with the Hessian matrix
H =
∂2π s
∂q 2∂ 2πs
∂q ∂rs
∂2π s
∂rs∂q ∂ 2πs
∂rs2
(A.1)
must be negative definite. Taking the second partial derivatives of (15) and (16),
we find that negative definiteness requires
∂ 2π s
∂q 2= −Nf ’(q )[q r + (1− q )rd − rs ] − (r − rd )Nf(q ) < 0 and (A.2)
∂ 2π s
∂rs2 = f ’(q’)
N(r − rd )2 [q ’r + (1− q’)rd − rs ] < 0. (A.3)
Given that the term in brackets of (A.2) is zero by (15) and that q’> q and r > rd ,
the term in brackets of (A.3) must be strictly positive. This implies that
f ’(q’ ) < 0. (A.4)
46
B. Proof of Proposition 1.
Securitization has no effect on equilibrium if q * < ˜ q , where ˜ q is determined by
setting (17) equal to (E.2), or q = q , which implies
rs (r ) = rf − δ . (B.1)
In (B.1), rs (r ) is described by (21) and (22). Let ˜ r be the value of r where (B.1)
holds. From (22), rs is increasing in r . It then follows that ˜ r is increasing in rf
and decreasing in δ : ˜ r (rf
+,δ
−) .32 Substituting ˜ r (rf
+,δ
−) in for r in (E.2), we obtain
˜ q =rf −δ − rd[ ]˜ r (rf
+,δ
−) − rd
. (B.2)
Equation (B.2) thus yields ˜ q (rf ,δ,rd ) . Using (6) and (B.2), we may write
q * < ˜ q ⇔ 1−(rb − rf )(rc − rd )
<rf −δ − rd[ ]˜ r (rf
+,δ
−) − rd
⇒ ˜ r (rf
+,δ
−)(rc − rd − rb + rf ) < rc (rf − δ) +δrd . (B.3)
From the last inequality in (B.3), we may bring rb and rc together on one side of
the inequality so that
q * < ˜ q ⇒ rf − rd −δrd
˜ r (rf
+,δ
−)
< rb − (1−α )rc , (B.4)
where α =rf − δ˜ r (rf
+,δ
−)
∈(0,1) because rs ≤ r ⇒ rf − δ < ˜ r . We may conclude that
combinations of rb and rc satisfying (B.4) are associated with equilibrium being
unaffected by securitization. In terms of proposition 1, (B.4) shows that a high
value of rb and/or low value of rc make the inequality more likely to hold.
32 We note that the value of ˜ r also depends on the shape of the pdf f(q).
47
Similarly, using the last inequality in (B.3), we may collect the terms
involving δ and write
q * < ˜ q ⇒δ(rd + rc ) − rcrf
˜ r (rf
+,δ
−)
< rb − rc + rd − rf . (B.5)
The second inequality in (B.5) is more likely to hold as δ becomes smaller.
Hence, we have shown that q * < ˜ q is more likely for larger values of rb , for
smaller values of rc , and for smaller values of δ .33 QED
33 The magnitudes ofrd and rf also affect the type of equilibrium achieved, but these effects
depend on the specific shape of f(q). For example, both q * and ˜ q are decreasing in rd , but the
magnitude of ∂ ˜ q ∂rd
, found by differentiating (B.2), depends on the gap between ˜ r and (rf − δ) ,
which we cannot solve for explicitly without knowing the functional form of f(q). Similarly, q * is
increasing in rf , but the effect of changes in rf on ˜ q is indeteminant.
48
C. Derivation that f > f 0 at the securitizer’s profit-maximizing choice of rs
Substitute (11) and (17) into (18). After rearranging, we get
f(q’) =F (q ’) −F (q )
q’−q (C.1)
Alternatively, consider the problem of choosing q’ to maximize F(q’)− F(q )
q ’−q given
the value of q . The first-order condition for this problem is
f(q’ )(q ’−q ) = F(q’ )− F(q ) , (C.2)
which is equivalent to (C.1), demonstrating that profit maximization implies
maximizing F(q’)− F(q )
q ’−q . In other words, we can think of the securitizer, given
q , choosing rs (and hence q’ ) to maximize the volume of securitization,
F(q’)− F(q ) , per unit of credit score range in the securitized pool, q’−q .
Let the profit-maximizing choice of q’ be denoted by q’* . We have
established that q’* maximizes F(q’)− F(q )
q ’−q at the point where
f(q’* ) =F(q’* ) −F(q )
q’* −q . (C.3)
Note that
f(q ) = limq ’→q
F (q ’) −F(q )q’−q
, (C.4)
which must be smaller than (C.3) because (C.3) yields the maximized value of
F(q’)− F(q )q ’−q
. Since (C.4) > (C.3), we have f(q’* ) > f (q ) , i.e.,
f > f 0 . (C.5)
QED